Year: 2026
Type: preprint
Abstract: We revisit the variational characterization of Nesterov's celebrated method and study when Nesterov trajectories are eventually locally minimizing in a variational sense, a notion that we define precisely. We show that this question is equivalent to the nonnegativity of an associated Schrödinger operator acting on variations of the trajectory. For power objectives of the form \[ f(x)=\frac{1}{p}\|x\|^p \] (where $\| \cdot\|$ denotes the Euclidean norm), we give a complete characterization: Among all exponents $p\ge 2$, the Nesterov trajectory with non-zero initial position is eventually locally minimizing if and only if $p=4$. We further interpret this exceptional exponent through the stability theory of semilinear elliptic equations.
Source: arXiv (Cornell University)
Authors: Morteza, Peyman, Wright, Stephen
Institution: University of Wisconsin–Madison
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223998043